A Random Integration Algorithm for High-dimensional Function Spaces

TL;DR

This paper presents a new random integration algorithm that achieves high accuracy and polynomial tractability in high-dimensional function spaces, especially for functions with sparse or rapidly decaying Fourier coefficients.

Contribution

The paper introduces a novel random integration algorithm that is nearly optimal and polynomially tractable across various high-dimensional function spaces without requiring prior weight knowledge.

Findings

Achieves nearly optimal RMSE bounds in Sobolev and Korobov spaces.

Ensures polynomial tractability, avoiding exponential sample growth with dimension.

Demonstrates effectiveness through numerical experiments.

Abstract

We introduce a novel random integration algorithm that boasts both high convergence order and polynomial tractability for functions characterized by sparse frequencies or rapidly decaying Fourier coefficients. Specifically, for integration in periodic isotropic Sobolev space and the isotropic Sobolev space with compact support, our approach attains a nearly optimal root mean square error (RMSE) bound. In contrast to previous nearly optimal algorithms, our method exhibits polynomial tractability, ensuring that the number of samples does not scale exponentially with increasing dimensions. Our integration algorithm also enjoys nearly optimal bound for weighted Korobov space. Furthermore, the algorithm can be applied without the need for prior knowledge of weights, distinguishing it from the component-by-component algorithm. For integration in the Wiener algebra, the sample complexity of our algorithm is independent of the decay rate of Fourier coefficients. The effectiveness of the integration is confirmed through numerical experiments.